Fundamental
Statistics for the Behavioral Sciences 7th
edition

David C.
Howell

Review of Arithmetic:

"I Don’t Do Fractions"

The subtitle tells it all. This was
actually the response I
received from a student in one of my classes—not a statistics
course, by the
way. As part of an in-class exercise, they had to convert simple
fractions to
decimals to calculate their average rating of the quality of chocolate
chip
cookies. But this student couldn’t handle the arithmetic, and
his friends had
to help him out.

This episode made me realize that there are some things
people like me take for granted, without remembering that there is a
whole world
out there filled with people who don’t use arithmetic on a
daily basis, and
have forgotten lots of stuff. I’m not talking about calculus
or differential
geometry, I’m talking about whether 1/2 + 1/3 = 1/5 or 1/6 or
5/6. (It equals
5/6.) So I have tried to put together a simple review of arithmetic,
with a
teenie-weenie bit of algebra thrown in for good measure. You may think
that some
of this material is too simple—I hope you do. But there is
probably something
here for everyone, since all of us forget even the simplest things when
we don’t
use them. So I’ll start out with what I hope everyone knows
without being
told, and progress toward some of the more troublesome topics. In each
case I
give the answer to one worked example, and then provide a few more
examples.

I am not a teacher of arithmetic, and some of the rules that
they use everyday, and can pass on to students, are not in my
head—they once
were, but I forget just like you do. That means that some of my
explanations may
be a bit clumsy, and may be of the form "well, it works for me." I
don’t
think there is anything wrong with that. We all develop our own way of
looking
at things, and perhaps my way will make more sense to you than the rule
you
learned in school; perhaps it won’t.

Addition and Subtraction

Everyone knows how to add, at least if all of the numbers are
positive, but I have to start somewhere. Most of the time we deal with
positive
numbers, because we want to talk about things like the cost of a
hamburger and a
coke. When we’re dealing with positive numbers we could affix
a + sign, but we
usually don’t.

(+5) + (+6) = 5 + 6 = 11

(I don’t think I need to supply test items for that!)

But remember, it doesn’t make any difference what
order the
items are in. If your hamburger costs $1.50, and your coffee costs
$.80, you don’t
really care about the order in which the clerk adds them up.

1.50 + .80 = .80 + 1.50 = $2.30

But what about negative numbers? Well, subtraction is really
just adding negative numbers, so I might as well cover them here?
Suppose that
you had a discount coupon worth $.25 that you used for your hamburger
and
coffee. Then you can just add that in too, affixing a minus sign.

1.50 + .80 + (-.25) = 1.50 + .80 – .25 = $2.05

So, to add negative numbers, you really just subtract the
negative numbers from the positive ones.

To make this a bit more complicated, assume that your girl
friend also had a coupon, though hers was worth $.40. You could either
just add
(and subtract) things as you go along, as in

1.50 + .80 + (-.25) + (-.40) = 1.50 + .80 - .25 - .40 = $1.65

or you can first collect terms having the same sign.

1.50 + .80 + (-.25) + (-.40) = 1.50 + .80 - .25 -.40 =

(1.50 + .80) – (.25 + .40) = 2.30 - .65 = $1.65

Notice that I added together the positive items (the food), I
added together the negative items (the coupons), and then I subtracted
the
negative sum from the positive sum.

Examples

2 – 1 + 5 – 3 – 4 + 8 = ?

5 + 4 – 3 – 2 + 4 – 2 = ?

12 + 14 – 6 + 3 – 10 = ?

What about 5 – 8 + 6? You probably don’t
have any trouble
with that, even though if you do it sequentially you have to go from 5
to –3
to +3. If those were much larger numbers, you could still do it the
same way,
but I prefer to collect terms with the same sign, add those two groups
up, and
then subtract the sum of the negative numbers from the sum of the
positive ones.
For example, add the 5 and 6 to get 11, and then subtract the 8 to get
3; i.e. 5
+ 6 = 11, 11 – 8 = 3.

But what if the negative numbers are larger than the positive
ones? For example, what about 23 – 36.

That’s easy, too. Just do the subtraction in
reverse, and
then reverse the sign. In other words, instead of subtracting 36 from
23,
subtract 23 from 36, but make the answer negative.

23 – 36 = –(36 – 23) =
–13.

Examples

8 – 9 = ?

17 – 22 = ?

54 – 87 =?

Parentheses

Parentheses are our friends. They make things so much
clearer, and it is very hard to go wrong with them if you follow a
simple rule.
Always work from the inside out!

One of the big advantages of parentheses is that they tell us
what order to do things in. For example, if you wanted to add 8 and 9,
and then
divide the answer by 3, you cannot write 8 + 9/3, because that would
tell you to
add 8 to the result of 9/3, which would be 11. Instead we put the
parentheses
around 8 + 9, and then show the division by 3, as in
(8 + 9)/3 = 17/3 = 5.667.

As an aside, you do the same thing when you are playing on
the Internet. If you want to look up information on George Washington
or Abe
Lincoln, you might type in "George and Washington or Abe and Lincoln."
However, your software, taking you very literally, would look for
documents that
contain all the names George and Washington and
Lincoln, or the name Abe.
That’s not what you wanted. You wanted (George and
Washington) or (Abe and
Lincoln). Notice how the parentheses keep things together that belong
together?
They do the same thing in arithmetic.

A big thing about parentheses is the way that they deal with
signs—particularly minus signs. In an earlier example I went
from 23 – 36 to
– (36 – 23). How did I do that? Well, I did it by
knowing that when I have a
negative outside a set of parentheses, and I remove the parentheses, I
just
reverse all the signs. For example,

–(23 –14 + 16 – 5 + 8 + 7) =
–23 + 14 – 16 + 5 – 8 – 7

This rule only indirectly answers the question I asked. I
asked how I could go from 23 – 36 to – (36
– 23), but I answered by going from –(36
– 23) to –36
+ 23 to 23 - 36. Well, that’s because I couldn’t
think of a rule to say it
the other way around. See if you can come up with one. The way I did it
works,
it’s just a bit clumsy.

About half way through the book you are going to come across
a formula like the following. z = ( – μ)/(σ /√N)
This formula doesn’t have any numbers in it, only symbols, so
it is really
algebra instead of arithmetic. But you interpret it the same way you
interpreted
the examples above. First you subtract μ
from .
Then you divide σ by the
square root of N.
Then you take the first answer and divide it by the second. All that I
am doing
is worrying about what comes inside parentheses before I worry about
what is
outside.

It gets even more fun.

The above example had two sets of parentheses. What if you
have one set nested within another? For example,

(8 + (5 + 1)/2)/7

You follow the same rule. Start inside the inner parentheses
and add 5 + 1 = 6. Then do the stuff within the next set of
parentheses, which tells you to divide 6 by 2 and add 8. This
gives you 8 + 3 = 11. Finally, divide that by 7, getting 1.57.

With nested parentheses we always work from the inside out.
That is true if we have truly nested parentheses, as in 4 × ((8 + 4)/3
– (6 + 3)/3) = 4 ×
(4 – 3)= 4(1) = 4, or if you use a mixture of parentheses and
brackets, as in 4 ×
[(8 + 4)/3 – (6 + 3)/3)]. Brackets, parentheses, and braces
"{ }" mean the same thing. We just sometimes mix them up so that our
eyes don’t get crossed trying to figure out which parenthesis
goes with (closes) which other parenthesis.

Try the following example. Note: treat a square root sign like
a set of parentheses—i.e. do whatever is under the square
root sign before you take the square root.

Example

How would you carry out the following steps, assuming that
you knew what the symbols meant?

Multiplication and Division

I show multiplication in several different ways in the text.
For example, I could write 2 ×
3, or (2)(3), or 2(3),
or 2*3. The latter is something that copy editors wrap my knuckles
over, but
some probably snuck through. They all mean the same thing.

Similarly in division, we can write 1/2, or ½, or
1÷2.
Again, they all mean the same thing, although the last one
isn’t used much
anymore.

I’ll start with some of the simplest rules.

Multiplication by 0 leaves 0 as an answer.

5 × 0
= 0

If you didn’t have any money yesterday, and you
have 5 times that much today, you’re still broke.

Multiplication by 1 doesn’t change anything.

1 × 5
= 5

You haven’t multiplied it by anything, so it
hasn’t gotten any bigger.

Division by 1 doesn't change anything.

5/1 = 5.

A $5 bill cut into one piece is still a $5 bill.

Division by zero is infinity, or undefined,
depending on
what you want to call it.

5/0 = ∞

If you had $5 yesterday and you chopped it up into 0
pieces, what have you got? Beats me! We try very hard never to get an
equation that works out to having 0 in the denominator, because that
brings us to a big halt.

Division of 0 by something is 0

0/5 = 0

If you didn’t have any money yesterday, and you
chopped that empty wallet into 5 pieces, it still doesn’t
have any money in it.(But that leaves the interesting question of what happens if you divide 0 by 0. Hmmm! There must be more important things to worry about.)

Multiplying positive and negative numbers.

This one is easy. When you multiply some positive numbers and
some negative numbers together, the answer is positive if there are an
even number of negative signs, and the answer is negative if there are
an odd number.

-2 ×
3 = -6

-2 ×
(-3) = 6

-2 ×
5 ×
(-8) × (-3) × 3 = -720

Examples

3 ×
(-5) = ?

-5 ×
(-8) × (-2 ) × 4 = ?

8 ×
(-2) × (-3) = ?

9 ×
(0/5) = ?

7/1 + 12 = ?

9 ×
(5/0) = ?

(-9 ×
12)/3 = ?

Dividing positive and negative numbers.

The logic here is the same as it was above. If there on an odd
number of negative signs, the answer is negative; otherwise it is
positive.

4/(-3) = -1.333

-4/3 = -1.333

-4/(-3) = 1.333

Examples

8/(-2) = ?

12/6 = ?

18/(-4) = ?

-15/3 = ?

-25/(-5) = ?

Fractions

Here is where students get confused. They know that there are
rules, but they don’t know what those rules are, so they give
up. I’ll
supply a few rules in a minute, but it is very much more important for
you to
realize that you can figure out what those rules are in about 15
seconds.

Before I go any further, however, I have to remind you about
numerators and denominators. The numerator is the thing on the top, and
the
denominator is the thing on the bottom. So in 5/7, the 5 is the
numerator and
the 7 is the denominator. You can remember this because the "de"
prefix is found in words like "denounce," "denigrate," and
"deficit," all of which have a connotation of "put down,"
which reminds you of putting something on the bottom.

If I gave you 1/5 + 2/5 and asked you what it was, you might
have a problem. (I hope you wouldn’t, but bear with me.) But
I’m sure that
you know that 1/3 plus 1/3 is 2/3. From that you ought to be able to
figure out
whether you are supposed to add the numerators, or add the
denominators, or add
both. The only way that you could get from 1/3 + 1/3 to 2/3 is to add
the
numerators but not the denominators. (If you added both numerator and
denominator, you would have 2/6, which is just 1/3, so that
can’t be right. If
you add just the denominators you would get 1/6, and that is less than
each of
the pieces, so it can’t be right either. That only leaves you
with the
possibility of adding the numerators, which does give the right answer.)

Adding simple fractions

The rule is that if two or more fractions have the same
denominator, you just add up the numerators and put that sum
over the common denominator. So

1/5 + 3/5 = 4/5.

1/8 + 2/8 + 3/8 = 6/8.

5/8 + 7/8 = 12/8 = 1 and 4/8 = 1.5

Examples

1/3 + 2/3 = ?

2/7 + 3/7 = ?

8/7 + 3/7 = ?

8/7 – 5/7 = ?

Notice that I snuck in a subtraction for the last one. You
should be able to figure out that if in addition you add the
numerators, in subtraction you must subtract them.

4/5 – 3/5 = 1/5

6/9 – 2/9 = 4/9

4/9 – 7/9 = -3/9

Examples

3/5 – 1/5 = ?

4/7 – 1/7 = ?

12/7 – 6/7 = ?

10/6 – 15/6 = ?

Adding messy fractions

Things get messy when you have different denominators. I just
said that if you have the same denominator, you can add (or subtract)
the numerators. But what if the denominators are different?

Well, we won’t let you have different denominators.
If they are different, you’ll just have to make them the same
before you go on. So, if you want to add 1/2 and 1/3, you’ll
have to find a way to express both of those with the same denominator.
Certainly you know that 1/2 = 3/6, and that 1/3 = 2/6, so just convert
the problem from 1/2 + 1/3 to 3/6 + 2/6 = 5/6

(Once you have the common denominator, then you can go ahead
and add the numerators.)

Examples

1/2 + 3/4 = ?

4/5 + 3/2 = ?

3/4 – 2/8 = ?

Adding really messy fractions

The system above works just fine if you have things like 3/4
and 2/3, but what about 7/22 + 3/17? Well, there is a rule for doing
that, but I’m not going to try to get you to use it, because
I can’t think when you would want to. The way that most
people do this is to convert each to a decimal, and then add the
decimal equivalents.

7/22 = .3182

3/17 = .1765

.4947

But how did you do that?

Here is where the student mentioned at the beginning of this
section fell apart. He apparently didn’t remember how to get
from 7/22 to .318. Well, you just have to remember that 7/22 really
means 7 divided by 22, and if you can’t do that division on
paper, you can certainly do it on a calculator. I’m not going
to try to teach you to do long division.

Examples

4/9 + 5/7 = ?

15/33 + 5/7 = ?

3/6 + 4/9 = ?

Multiplying fractions

When you multiply fractions, you just multiple the numerators
together and the denominators together.

1/2 ×
1/2 = 1/4 , which is what you would expect.

2/3 ×
4/5 = 8/15

7/3 ×
6/2 = 42/6 = 7

Examples

2/3 ×
1/2 = ?

4/5 ×
3/4 = ?

7/8 ×
6/13 = ?

Notice that in each of the answers I have "reduced the
fraction" when I could. Since both the numerator and the denominator in
2/8 can be divided by 2, I can do that to get 2/8 = 1/4. It
doesn’t hurt if you don’t, but it is much neater if
you do. (Who wants to look at 21/49 when they can divide both by 7 and
get 3/7?)

Multiplying signed fractions

What if we have fractions with minus signs attached? For
example

-3/4 ×
2/5 = -6/20 = -3/10

You just do exactly the same thing as you do when you multiply
whole numbers. If the number of minus signs are even, the answer is
positive. If the number of minus signs are odd, the answer is negative.

Examples

-3/5 ×
2/3 = ?

-3/7 ×
-4/9 = ?

-1/3 ×
3/4 × -3/8 = ?

-1/2 ×
2/3 × 4/5 = ?

Multiplying a fraction by a whole number

What about 2 ×
4/5? Well, that is really just 2/1 ×
4/5, so you multiply the numerators and multiply the denominators,
which gives (2 ×
4)/(1 × 5) =
8/5, which is the same as just multiplying the numerator by the whole
number.

What if the multiplier is a decimal?

Nothing changes. Just multiply the numerator by the decimal.

0.4 ×
3/7 = (0.4 ×
3)/7 = (1.2)/7

Dividing fractions

Here things could get a bit messier. What about (2/3) / (3/4)?

There are a couple of ways you could do this. You could start
by converting each fraction to a decimal, and then dividing the two
decimals.

(2/3) / (3/4) = 0.6667 / 0.75 = 0.4444

That will always work, but it is bit on the overkill side if
you have nice tidy fractions. What I do is to mentally invert the
denominator, and then multiply. (By "invert the denominator" I mean
"flip it upside down."

That will work with even messy fractions, but for messy
fractions it is easier to get the decimal equivalents first.

Examples

(1/3) / (2/3) = ?

(5/6) / (8/7) = ?

(12/7) / (7/39) = ?

Algebraic fractions

The same rule that applies to numerical fractions also applies
to algebraic ones. For a one-sample t test we are
going to have something like

Notice that I have just inverted the denominator and
multiplied it by the numerator. There are very few times when you would
have to do something like that with this text, but I thought I would
sneak in a little aside. Notice that you don’t even have to
know what any of those symbols in the equation represents to be able to
do the manipulation.

Decimals and Rounding

Many of the numbers we will deal with are decimals, and we
don’t usually want to carry around all of those decimal
places. This leaves us
with two questions. How many decimals should we carry, and how should
we round
up or down to them.

Decimals

There are all sorts of rules about how many decimal places
to carry, and whatever I say will upset someone. So I’ll just
go ahead and give an answer without trying to defend it. (I’d
probably have a hard time defending it to a mathematical purist.)

If the numbers you are working with are whole numbers, carry
2 or three decimal places. Thus represent 234.6778767 as 234.68 or
maybe 234.678

If you have very small numbers, keep 1 to 3 non-zero
decimals. Thus represent .00044178 as .0004 or .00044.

There are lots of places in the book where I violate these
rules, sometimes for good reasons, and sometimes because I am careless.
But the rules are at least rough guidelines to go by.

Rounding

There is not only continual dispute over how many decimal
places to carry, but also over how to cut down to that many decimal
places. I am going to state a very simple set of rules.

If you are more than half way toward the next number,
round up.

If you are less than half way toward the next number,
round down.

If you are teetering one the exact center, make the result
even.

Suppose that we have the number 2567.45654 and we want to
round it to a whole number. You are somewhere between 2567 and 2568,
and half way would be 2567.500000000. So you are less than half way to
2568, so round down to 2567..

But suppose you had 2567.656782. Here you are more than half
way toward 2568, so round up. I would do this even if you had
2567.50002—It is still more than half way to 2568.

Now suppose that you had 2567.500000. Here you are exactly
half way in the middle. You aren’t any closer to 2567 than to
2568. My rule is to make the result even, so I would round up to 2568.
But, if I had been at 2566.50000, I would have rounded down to 2566 to
make the result even.

The only time when you try this odd/even trick is when you
are exactly on the border. If you are closer to one end or the other,
go to that end. The odd/even bit means that you will round up about
half the time and round down about half the time. What could be fairer?

The same rules apply no matter how many decimal places you
chose to have. So, if you want to carry 3 decimals, then

If you have a slightly more complex expression, such as 5X2,
you square Xbefore you
multiply by 5.

Finally, raising something to some other power is the same
basic idea. The cube of a number is that number raised to the third
power. So X3
= X ×
X ×
X.

Examples

62 = ?

8(32) = ?

0.52 = ?

Squares of expressions

Remember the rule about parentheses that I gave you earlier.
If you have an expression like (5 + 12)2, you do
the stuff within the parentheses before you do the stuff outside the
parentheses. So you first add 5 and 12 to get 17, and then you square
17, which is 172 = 289. It is definitely not
52 + 122 = 25 + 144 =
169. Always do what is within parentheses before you do
anything else.

Square roots

The square root of the number is just that value which, when
squared, equals the number. For example, we know that 52
= 25, so
The thing over the 25 is called a radical, and is a square root symbol.
(Unless
you have an equation editor, it is hard to make a proper radical on a
word
processor. The closest we can come is the √ symbol. When you
see √25.7,
it means the same as —you
just have to pretend that the bar extends over the digits.)

Once upon a time, when I was a little boy in about 7th
grade, I was taught how to take the square root of any number with
pencil and
paper. You set the problem up sort of like a long division problem, did
a bit of
magic, and there was your answer. I was pretty good at everything but
the magic
bit, but that I never could work out. Well, you’re in luck.
You have a
calculator, and it has a square root key, and that’s how we
find square roots
these days.

Technically, 25 actually has 2 square roots. 52
=
25, and (-5)2 = 25, but in statistics we always
use the positive
square root of a number.

Just as we do things within parentheses, and then square the
result, when getting square roots we do what is under the radical
before we take
the square root. So,
We do the addition first.

Percentages and Proportions

Percentages and proportions are very similar, and sometimes
get people confused. In fact, a percentage is really just a proportion
times
100. If your class has 32 students and 29 did well on the last exam,
then the
proportion of those doing well is 29/32 = .91. That means that 91% of
the
students did well.

To make things even messier, we sometimes express a
probability as a percentage—which is really bad. A
probability is a number
between 0 and 1. I can say that the probability of rain is .35, but I
should
never say that it is 35%. Of course, I do say just that on occasion,
but that
doesn’t make it right.

Don’t let yourself get confused between percentage
and
percentage points. Suppose that for your class of 32 students, 25 did
well on
the first exam and 29 did well on the second. Converting to percentages
we have
78% and 91% doing well on the two exams. The percentage of students
doing well
increase 13 percentage points, from 79% to 91%. But
that is a (29 –
25)/25 = 4/25 = 16 percent increase in the number
of students doing well.

Inequalities

You will frequently come across inequalities in this book.
They are things like less than (<), less than or equal to (<),
greater than (>), and greater than or equal to (>).
The most
frequent use of an inequality that you will see is the expression p
<
.05, which translates to "the probability is less than .05." Sometimes
you will see p > .05, which obviously means
that the probability is
now greater than .05.

Another of the expressions you will see is something of the
form 25 <X<
75. This really means that some
quantity we call X is between 25 and 75. (It is
read "25 is less
than or equal to X, which is less than or equal to
75." When you see
multiple sets of < like this, just read it
as "between."

Absolute values

I will occasionally refer to absolute values, symbolized by
something like |4|. This simply means to ignore whatever sign there is
for 4,
which is basically the same as saying "treat 4 as positive."

|4| = |-4| = 4

|6-9| = 3

The most common way in which you will see absolute values is
in the statement.
|X – |.
This says to subtract
from X, throw away the minus sign if there is one, and treat the answer
as
positive.

You will also come across a statement like p(|z|
) > 1.96 = .05. This translates to "the probability that the
absolute
value of z will be greater than 1.96 is .05." That,
itself,
translates to "the probability is .05 that z will
be more positive
than 1.96 or more negative than –1.96."

Equations

We have already been working with equations, but I should put
a few things in a section devoted to equations. The material here is
material
that you can expect to see in the text. I am leaving out topics that I
don’t
think you are likely to come across.

An equation is an expression with an equal sign separating
two quantities. Thus

Y = bX + a

is an equation that says that the thing on the left (Y
) is equal to the
expression on the right. You probably already knew that. But what about
a few
things that you might have forgotten?

If you do the same thing to both sides of the equation, you
haven’t really changed anything.

For example, you can add 12.6 to both sides of the equation,
and the equality is still true.

Y + 12.6 = bX + a
+ 12.6

If you can add, you can also subtract, because subtraction is
just addition of a negative number.

You can multiply (or divide) both sides of the equation by the
same thing without changing the equality.

3 ×
Y = 3(bX + a)

That basic rule tells us a lot about how to solve equations.

For example, suppose that we have

3Y = 7X + 8.

If we divide both sides by 3 we have

3Y/3 = (7X +8)/3

Therefore

Y = (7X +8)/3

In my own head I think of it slightly differently. If there
is a multiplier on one side of the equation, I move it to the other
side as a divisor, Similarly, if there is a divisor on one side, I
think of moving it to the other side as a multiplier. But I am doing
exactly what this rule says, I’m just not thinking about it
in that way.

A similar rule applies to addition and subtraction. Take the
same example:

3Y = 7X + 8.

3Y – 8 = 7X

I just moved the 8 to the other side and made it negative.
Or, if you prefer the more proper interpretation, I just subtracted 8
from both sides of the equation.

Now let’s put the two together.

If I want to solve for X, I must get rid
of that 8. I did that in the last step. Then I get rid of the 7 by
dividing.

3Y = 7X + 8.

3Y – 8 = 7X

(3Y –8)/7 = X

One more complication.

Suppose that we have one side of an equation squared. For
example, you will see expressions for the variance and the standard
deviation, one of which is just the square of the other.

The variance is

The standard deviation is the square root of the variance, so
we just take the square root of both sides. The square root of s2
is s, and will just put a radical around the right side to show that as
a square root.

(Radicals are a nuisance to deal with, so you will often see
statisticians square both sides of the equation before carrying out
other operations.

Conclusion

I have covered the very basic material here, but what I
covered should prepare you for just about anything you will see in the
book. By
far the most important thing to take away from this section is the idea
that if
you don’t know how to do something, make up an example with
very simple
numbers, and play with it. When you figure out how to make the simple
numbers
come out right, you will have also figured out how to make the more
complicated
stuff come out right.

There is another way that is really important. ASK GOOGLE!! Just type in "How do I multiply fractions?" and I guarantee that you will get at least once answer that is sensible and easy to understand. Don't ever forget that Google is your friend!!

If there is something that I have left out, please let me
know. Web sites can be updated daily, and it is easy to do.

Finally, if you want more review, try the web. There is a
very good site at http://www.icmicroanalysis.com/mathtutorials.htm . Just scroll to the middle of the page and there are a bunch of good links--though the level of difficult varies.